Problem 55 - Elementary and Intermediate Algebra

Question

Determine whether the given number is a solution of the given equation. $$ 15 ; x+17=32 $$

Step-by-Step Solution

Answer

Yes, 15 is a solution of the equation.

Step 1: Identify the equation and the number

The equation provided is \(x + 17 = 32\) and the number to check is 15.

Step 2: Substitute the number into the equation

Replace \(x\) with 15 in the equation: \(15 + 17 = 32\).

Step 3: Simplify the left-hand side

Add 15 and 17 together to get 32: \(15 + 17 = 32\).

Step 4: Compare the sides of the equation

The resulting equation is \(32 = 32\), which is true.

Step 5: Conclusion

Since substituting 15 into the equation resulted in a true statement, 15 is a solution of the equation \(x + 17 = 32\).

Key Concepts

substitution method

The substitution method is an effective technique for solving linear equations. It involves replacing a variable with a given number to determine if it satisfies the equation. In our example, we are given the equation \(x + 17 = 32\) and asked to check if the number 15 is a solution. To use the substitution method, follow these steps:

First, replace the variable \(x\) with 15. This substitution transforms the equation to \(15 + 17 = 32\).

Then, perform the necessary arithmetic operations to simplify the left side of the equation. If both sides of the equation are equal after substitution, the given number is indeed a solution.

equation verification

Equation verification is the process of confirming whether a specific value satisfies a given equation. Let's see how we verify our equation using 15:

First, substitute 15 into the equation \(x + 17 = 32\), resulting in \(15 + 17 = 32\).

Next, add the numbers on the left-hand side: \(15 + 17\). This results in 32. Thus, our new equation is \(32 = 32\).

Since both sides of the resulting equation are equal, the statement is true, verifying that 15 is indeed a solution. This process can be applied to any linear equation to test potential solutions.

arithmetic operations

Arithmetic operations involve basic mathematical processes such as addition, subtraction, multiplication, and division. They are fundamental when working with equations. Let's consider our example equation \(x + 17 = 32\) to demonstrate these operations:

After substituting \(x\) with 15, we need to perform addition: \(15 + 17\).
By adding 15 and 17, we get 32.

This simplification shows that the left-hand side of the equation equals the right-hand side. Understanding and correctly performing these operations is crucial for solving and verifying equations effectively.